Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6668044 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #2407 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1475 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,7)(4,6)(5,8)(17,18,23,27,22,26,21,28)(19,24,36,38,30,40,31,39)(20,32,33,25,35,42,34,41)(29,43,45,37,44,48,46,47);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,16)(13,14)(17,19)(18,24)(20,29)(21,31)(22,30)(23,36)(25,37)(26,40)(27,38)(28,39)(32,43)(33,45)(34,46)(35,44)(41,47)(42,48);(10,14)(15,16)(17,20)(18,25)(19,29)(21,33)(22,35)(23,34)(24,37)(26,41)(27,32)(28,42)(30,44)(31,45)(36,46)(38,43)(39,48)(40,47);(2,6)(5,7)(9,11,13,12)(10,15,14,16)(18,26)(19,30)(25,41)(27,28)(29,44)(31,36)(32,42)(45,46);(3,8)(5,7)(9,12,13,11)(10,15,14,16)(17,21,22,23)(18,27,26,28)(19,31,30,36)(20,33,35,34)(24,38,40,39)(25,32,41,42)(29,45,44,46)(37,43,47,48)Polycyclic presentation:
2^9|a^2=gi,d^2=h,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=dfi,e^a=ei,c^b=ch,d^b=di,e^b=efh,g^c=gi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6774998);